As usual, DEPENDS!
If you're mechanical and practiced, use a short straight takeback, and want to hit with topspin, maybe YES!
If you're athletic and a natural, use loopy takebacks, and what to hit with topspin, then the answer is FOR A MOMENT IN TIME!
Somewhere, somehow, someplace, sometime, for a true heavy topspin 2HBH, the racketface most close somewhat...

Do you get your racket in the pat the dog position for both one handed backhand and double handed backhand?

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You do not get your racquet in the pat-the-dog position for the onehanded backhand. For the two handed backhand you can. I won't go as far as saying you should, however, it doesn't hurt either.

When I teach the forehand and the twohander, I teach the same four positions for them. It makes it extremely easy to learn both strokes in a very short amount of time. However, I am not as critical with the "pat-the-dog" position on the twohander. Some players wrists/hands can not flatten out the head that much as they do in the forehand. So, I look for a slant in the racquet head that angles toward the pat-the-dog position.

The onehanded backhand is performed differently, so I do not use the pat-the-dog method with this stroke. The main issue with the onehanded backhand that needs to be engrained right away is making sure you understand how the "smile" pattern works and drill it in your skull. Besides your footwork, balance, and positioning for the onehanded backhand, the hand path in your backswing and forward swing is thee most important thing to engrain.

As normal, I'm the pointer outer of useless information.
JohnMcEnroe often held his racket SLIGHTLY closed to hit lower incoming balls with topspin. I think he was about the only top player who did this. He used the extremely short prep idea on his groundies.
Not saying anyone CAN or SHOULD copy anything about McEnroe's game, it's a gift recieved by maybe only him in the whole world.
I try it when my topspin backhand goes astray or for return of faster serves (close the racketface on the short backswing).